Pricing of forwards contracts

Question

Of the courses I am taking in college this semester, two are Financial Mathematics and Derivatives. In each course, we learn different formulas to calculate the forward price of a forward contract. Obviously, the two formulas must equal each other, but I am not sure what the link is i.e. how to get from one formula to the other.

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In my Derivatives course, I was taught that $$F_0 = S_0 e^{(r - d)T},$$ where

$F_0$ is the forward price,

$S_0$ is the spot price,

$r$ is the risk-free rate,

$d$ is the average yield per annum and

$T$ is the length of the contract (in years).

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In my Financial Mathematics course, it is also given that $$F_0 = (S_0 - PV_I) e^{\delta T},$$ where

$F_0$ is the forward price,

$S_0$ is the spot price,

$PV_I$ is the present value of the fixed income payment(s) due during the term of the contract,

$\delta$ is the force of interest and

$T$ is the length of the contract (in years).

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How do I reconcile the two formulas? In other words, how can I prove that $$F_0 = S_0 e^{(r - d)T} = (S_0 - PV_I) e^{\delta T}?$$

Any intuitive explanations will be greatly appreciated! :)

Answer 1

Decompose the first formula as $F_0=(S_0 - S_0(1-e^{-dT}))e^{rT}$ then let $PV_{I} = S_0(1-e^{-dT})$ which represents the present value of dividends (dividend rate = $d$) paid on the security during the life of the contract, and you obtain the second formula.